Theorem ptcmpg 21861

Description: Tychonoff's theorem: The product of compact spaces is compact. The choice principles needed are encoded in the last hypothesis: the base set of the product must be well-orderable and satisfy the ultrafilter lemma. Both these assumptions are satisfied if ~P ~P X is well-orderable, so if we assume the Axiom of Choice we can eliminate them (see ptcmp 21862). (Contributed by Mario Carneiro, 27-Aug-2015.)

Hypotheses

Ref	Expression
ptcmpg.1	|- J = ( Xt_ ` F )
ptcmpg.2	|- X = U. J

Assertion

Ref	Expression
ptcmpg	|- ( ( A e. V /\ F : A --> Comp /\ X e. (UFL i^i dom card ) ) -> J e. Comp )

Proof of Theorem ptcmpg

Dummy variables a b k m n u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptcmpg.1	. 2 |- J = ( Xt_ ` F )
2		nfcv 2764	. . . 4 |- F/_ k ( F ` a )
3		nfcv 2764	. . . 4 |- F/_ a ( F ` k )
4		nfcv 2764	. . . 4 |- F/_ k ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` a ) ) " b )
5		nfcv 2764	. . . 4 |- F/_ u ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` a ) ) " b )
6		nfcv 2764	. . . 4 |- F/_ a ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` k ) ) " u )
7		nfcv 2764	. . . 4 |- F/_ b ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` k ) ) " u )
8		fveq2 6191	. . . 4 |- ( a = k -> ( F ` a ) = ( F ` k ) )
9		fveq2 6191	. . . . . . . 8 |- ( a = k -> ( w ` a ) = ( w ` k ) )
10	9	mpteq2dv 4745	. . . . . . 7 |- ( a = k -> ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` a ) ) = ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` k ) ) )
11	10	cnveqd 5298	. . . . . 6 |- ( a = k -> `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` a ) ) = `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` k ) ) )
12	11	imaeq1d 5465	. . . . 5 |- ( a = k -> ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` a ) ) " b ) = ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` k ) ) " b ) )
13		imaeq2 5462	. . . . 5 |- ( b = u -> ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` k ) ) " b ) = ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` k ) ) " u ) )
14	12, 13	sylan9eq 2676	. . . 4 |- ( ( a = k /\ b = u ) -> ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` a ) ) " b ) = ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` k ) ) " u ) )
15	2, 3, 4, 5, 6, 7, 8, 14	cbvmpt2x 6733	. . 3 |- ( a e. A , b e. ( F ` a ) |-> ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` a ) ) " b ) ) = ( k e. A , u e. ( F ` k ) |-> ( `' ( w e. X_ n e. A U. ( F ` n ) |-> ( w ` k ) ) " u ) )
16		fveq2 6191	. . . . 5 |- ( n = m -> ( F ` n ) = ( F ` m ) )
17	16	unieqd 4446	. . . 4 |- ( n = m -> U. ( F ` n ) = U. ( F ` m ) )
18	17	cbvixpv 7926	. . 3 |- X_ n e. A U. ( F ` n ) = X_ m e. A U. ( F ` m )
19		simp1 1061	. . 3 |- ( ( A e. V /\ F : A --> Comp /\ X e. (UFL i^i dom card ) ) -> A e. V )
20		simp2 1062	. . 3 |- ( ( A e. V /\ F : A --> Comp /\ X e. (UFL i^i dom card ) ) -> F : A --> Comp )
21		cmptop 21198	. . . . . . . 8 |- ( k e. Comp -> k e. Top )
22	21	ssriv 3607	. . . . . . 7 |- Comp C_ Top
23		fss 6056	. . . . . . 7 |- ( ( F : A --> Comp /\ Comp C_ Top ) -> F : A --> Top )
24	20, 22, 23	sylancl 694	. . . . . 6 |- ( ( A e. V /\ F : A --> Comp /\ X e. (UFL i^i dom card ) ) -> F : A --> Top )
25	1	ptuni 21397	. . . . . 6 |- ( ( A e. V /\ F : A --> Top ) -> X_ n e. A U. ( F ` n ) = U. J )
26	19, 24, 25	syl2anc 693	. . . . 5 |- ( ( A e. V /\ F : A --> Comp /\ X e. (UFL i^i dom card ) ) -> X_ n e. A U. ( F ` n ) = U. J )
27		ptcmpg.2	. . . . 5 |- X = U. J
28	26, 27	syl6eqr 2674	. . . 4 |- ( ( A e. V /\ F : A --> Comp /\ X e. (UFL i^i dom card ) ) -> X_ n e. A U. ( F ` n ) = X )
29		simp3 1063	. . . 4 |- ( ( A e. V /\ F : A --> Comp /\ X e. (UFL i^i dom card ) ) -> X e. (UFL i^i dom card ) )
30	28, 29	eqeltrd 2701	. . 3 |- ( ( A e. V /\ F : A --> Comp /\ X e. (UFL i^i dom card ) ) -> X_ n e. A U. ( F ` n ) e. (UFL i^i dom card ) )
31	15, 18, 19, 20, 30	ptcmplem5 21860	. 2 |- ( ( A e. V /\ F : A --> Comp /\ X e. (UFL i^i dom card ) ) -> ( Xt_ ` F ) e. Comp )
32	1, 31	syl5eqel 2705	1 |- ( ( A e. V /\ F : A --> Comp /\ X e. (UFL i^i dom card ) ) -> J e. Comp )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   U.cuni 4436   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   "cima 5117   -->wf 5884   ` cfv 5888   |-> cmpt2 6652   X_cixp 7908   cardccrd 8761   Xt_cpt 16099   Topctop 20698   Compccmp 21189  UFLcufl 21704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-wdom 8464  df-card 8765  df-acn 8768  df-topgen 16104  df-pt 16105  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cmp 21190  df-fil 21650  df-ufil 21705  df-ufl 21706  df-flim 21743  df-fcls 21745

This theorem is referenced by:  ptcmp  21862  dfac21  37636